Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
4th Edition
ISBN: 9781337687805
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
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Chapter 4.7, Problem 26E

(a) Use Newton’s method with x1 = 1 to find the root of the equation x3 − x = 1 correct to six decimal places.

(b) Solve the equation in part (a) using x1 = 0.6 as the initial approximation.

(c) Solve the equation in part (a) using x1 = 0.57.

(You definitely need a programmable calculator for this part.)

(d) Graph f(x) = x3 − x − 1 and its tangent lines at x1 = 1, 0.6, and 0.57 to explain why Newton’s method is so sensitive to the value of the initial approximation.

(a)

Expert Solution
Check Mark
To determine

To find: The root of the equation x3−x=1 correct to six decimal places.

Answer to Problem 26E

The root is 1.324718.

Explanation of Solution

Formula used:

Newton’s method

xn+1=xn−f(xn)f′(xn)

Given:

f(x)=x3−x−1 with x1=1

Calculation:

f(x)=x3−x−1

Differentiate with respect to x ,

f′(x)=3x2−1

xn+1=xn−f(xn)f′(xn)=xn−xn3−xn−13xn2−1

x1=1

Substitute n=1 in xn+1=xn−xn3−xn−13xn2−1

x2=x1−x13−x1−13x12−1

Substitute x1=1 in x2=x1−x13−x1−13x12−1

x2=1−13−1−13×12−1=1−−12=1+12=1.5

Substitute n=2 in xn+1=xn−xn3−xn−13xn2−1

x3=x2−x23−x2−13x22−1

Substitute x2=1.5 in x2=x1−x13−x1−13x12−1

x3=1.5−1.53−1.5−13×1.52−1=1.347826

Substitute n=3 in xn+1=xn−xn3−xn−13xn2−1

x4=x3−x33−x3−13x32−1

Substitute x3=1.347826 in x4=x3−x33−x3−13x32−1

x4=1.347826−1.3478263−1.347826−13×1.3478262−1=1.325200

Substitute n=4 in xn+1=xn−xn3−xn−13xn2−1

x5=x4−x43−x4−13x42−1

Substitute x4=1.325200 in x5=x4−x43−x4−13x42−1

x5=1.325200−1.3252003−1.325200−13×1.3252002−1=1.324718

Substitute n=5 in xn+1=xn−xn3−xn−13xn2−1

x6=x5−x53−x5−13x52−1

Substitute x5=1.324718 in x6=x5−x53−x5−13x52−1

x6=1.324718−1.3247183−1.324718−13×1.3247182−1=1.324718

x5=x6

Hence the root is 1.324718

(b)

Expert Solution
Check Mark
To determine

To find: The root of the equation x3−x=1 correct to six decimal places.

Answer to Problem 26E

The root is 1.324718.

Explanation of Solution

Given:

f(x)=x3−x−1 with x1=1

Formula used:

Newton’s method

xn+1=xn−f(xn)f′(xn) 

Calculation:

f(x)=x3−x−1

Differentiate with respect to x ,

f′(x)=3x2−1

xn+1=xn−f(xn)f′(xn)=xn−xn3−xn−13xn2−1

x1=0.6

Substitute n=1 in xn+1=xn−xn3−xn−13xn2−1

x2=x1−x13−x1−13x12−1

Substitute x1=1 in x2=x1−x13−x1−13x12−1

x2=0.6−0.63−0.6−13×0.62−1=17.9

Substitute n=2 in xn+1=xn−xn3−xn−13xn2−1

x3=x2−x23−x2−13x22−1

Substitute x2=17.9 in x2=x1−x13−x1−13x12−1

x3=17.9−17.93−17.9−13×17.92−1=11.946802

Substitute n=3 in xn+1=xn−xn3−xn−13xn2−1

x4=x3−x33−x3−13x32−1

Substitute x3=11.946802 in x4=x3−x33−x3−13x32−1

x4=11.946802−11.9468023−11.946802−13×11.9468022−1=7.985520

Substitute n=4 in xn+1=xn−xn3−xn−13xn2−1

x5=x4−x43−x4−13x42−1

Substitute x4=7.985520 in x5=x4−x43−x4−13x42−1

x5=7.985520−7.9855203−7.985520−13×7.9855202−1=5.356909

Substitute n=5 in xn+1=xn−xn3−xn−13xn2−1

x6=x5−x53−x5−13x52−1

Substitute x5=5.356909 in x6=x5−x53−x5−13x52−1

x6=5.356909−5.3569093−5.356909−13×5.3569092−1=3.624996

Substitute n=6 in xn+1=xn−xn3−xn−13xn2−1

x7=x6−x63−x6−13x62−1

Substitute x6=3.624996 in x7=x6−x63−x6−13x62−1

x7=3.624996−3.6249963−3.624996−13×3.6249962−1=2.505589

Substitute n=7 in xn+1=xn−xn3−xn−13xn2−1

x8=x7−x73−x7−13x72−1

Substitute x7=2.505589 in x8=x7−x73−x7−13x72−1

x8=2.505589−2.5055893−2.505589−13×2.5055892−1=1.820129

Substitute n=8 in xn+1=xn−xn3−xn−13xn2−1

x9=x8−x83−x8−13x82−1

Substitute x8=1.820129 in x9=x8−x83−x8−13x82−1

x9=1.820129−1.8201293−1.820129−13×1.8201292−1=1.461044

Substitute n=9 in xn+1=xn−xn3−xn−13xn2−1

x10=x9−x93−x9−13x92−1

Substitute x9=1.461044 in x10=x9−x93−x9−13x92−1

x10=1.461044−1.4610443−1.461044−13×1.4610442−1=1.339323

Substitute n=10 in xn+1=xn−xn3−xn−13xn2−1

x11=x10−x103−x10−13x102−1

Substitute x10=1.339323 in x11=x10−x103−x10−13x102−1

x11=1.339323−1.3393233−1.339323−13×1.3393232−1=1.324913

Substitute n=11 in xn+1=xn−xn3−xn−13xn2−1

x12=x11−x113−x11−13x112−1

Substitute x11=1.324913 in x12=x11−x113−x11−13x112−1

x12=1.324913−1.3249133−1.324913−13×1.3249132−1=1.324718

Substitute n=12 in xn+1=xn−xn3−xn−13xn2−1

x13=x12−x123−x12−13x122−1

Substitute x12=1.324718 in x13=x12−x123−x12−13x122−1

x13=1.324718−1.3247183−1.324718−13×1.3247182−1=1.324718

x12=x13

Hence the root is 1.324718

(c)

Expert Solution
Check Mark
To determine

To find: The root of the equation x3−x=1 correct to six decimal places.

Answer to Problem 26E

The root is 1.324718

Explanation of Solution

Given:

f(x)=x3−x−1 with x1=1

Formula used:

Newton’s method

xn+1=xn−f(xn)f′(xn)

Calculation:

f(x)=x3−x−1

Differentiate with respect to x ,

f′(x)=3x2−1

xn+1=xn−f(xn)f′(xn)=xn−xn3−xn−13xn2−1

x1=0.57

Substitute n=1 in xn+1=xn−xn3−xn−13xn2−1

x2=x1−x13−x1−13x12−1

Substitute x1=0.57 in x2=x1−x13−x1−13x12−1

x2=0.57−0.573−0.57−13×0.572−1=−54.165455

Substitute n=2 in xn+1=xn−xn3−xn−13xn2−1

x3=x2−x23−x2−13x22−1

Substitute x2=−54.165455 in x2=x1−x13−x1−13x12−1

x3=−54.165455−−54.1654553+54.165455−13×−54.1654552−1=−36.114293

Substitute n=3 in xn+1=xn−xn3−xn−13xn2−1

x4=x3−x33−x3−13x32−1

Substitute x3=−36.114293 in x4=x3−x33−x3−13x32−1

x4=−36.114293−−36.1142933+36.114293−13×−36.1142932−1=−24.082094

Substitute n=4 in xn+1=xn−xn3−xn−13xn2−1

x5=x4−x43−x4−13x42−1

Substitute x4=−24.082094 in x5=x4−x43−x4−13x42−1

x5=−24.082094−−24.0820943+24.082094−13×−24.0820942−1=−16.063387

Substitute n=5 in xn+1=xn−xn3−xn−13xn2−1

x6=x5−x53−x5−13x52−1

Substitute x5=−16.063387 in x6=x5−x53−x5−13x52−1

x6=−16.063387−−16.0633873+16.063387−13×−16.0633872−1=−10.721483

Substitute n=6 in xn+1=xn−xn3−xn−13xn2−1

x7=x6−x63−x6−13x62−1

Substitute x6=−10.721483 in x7=x6−x63−x6−13x62−1

x7=−10.721483−−10.7214833+10.721483−13×−10.7214832−1=−7.165534

Substitute n=7 in xn+1=xn−xn3−xn−13xn2−1

x8=x7−x73−x7−13x72−1

Substitute x7=−7.165534 in x8=x7−x73−x7−13x72−1

x8=−7.165534−−7.1655343+7.1655349−13×−7.1655342−1=−4.801704

Substitute n=8 in xn+1=xn−xn3−xn−13xn2−1

x9=x8−x83−x8−13x82−1

Substitute x8=−4.801704 in x9=x8−x83−x8−13x82−1

x9=−4.801704−−4.8017043+4.801704−13×−4.8017042−1=−3.233425

Substitute n=9 in xn+1=xn−xn3−xn−13xn2−1

x10=x9−x93−x9−13x92−1

Substitute x9=−3.233425 in x10=x9−x93−x9−13x92−1

x10=−3.233425−−3.2334253+3.233425−13×−3.2334252−1=−2.193674

Substitute n=10 in xn+1=xn−xn3−xn−13xn2−1

x11=x10−x103−x10−13x102−1

Substitute x10=−2.193674 in x11=x10−x103−x10−13x102−1

x11=−2.193674−−2.1936743+2.193674−13×−2.1936742−1=−1.496867

Substitute n=11 in xn+1=xn−xn3−xn−13xn2−1

x12=x11−x113−x11−13x112−1

Substitute x11=−1.496867 in x12=x11−x113−x11−13x112−1

x12=−1.496867−−1.4968673+1.496867−13×−1.4968672−1=−0.997546

Substitute n=12 in xn+1=xn−xn3−xn−13xn2−1

x13=x12−x123−x12−13x122−1

Substitute x12=−0.997546 in x13=x12−x123−x12−13x122−1

x13=−0.997546−−0.9975463+0.997546−13×−0.9975462−1=−0.496305

Substitute n=13 in xn+1=xn−xn3−xn−13xn2−1

x14=x13−x133−x13−13x132−1

Substitute x13=−0.496305 in x14=x13−x133−x13−13x132−1

x14=−0.496305−−0.4963053+0.496305−13×−0.4963052−1=−2.894162

Substitute n=14 in xn+1=xn−xn3−xn−13xn2−1

x15=x14−x143−x14−13x142−1

Substitute x14=−2.894162 in x15=x14−x143−x14−13x142−1

x15=−2.894162−−2.8941623+2.894162−13×−2.8941622−1=−1.967962

Substitute n=15 in xn+1=xn−xn3−xn−13xn2−1

x16=x15−x153−x15−13x152−1

Substitute x15=−1.967962 in x16=x15−x153−x15−13x152−1

x16=−1.967962−−1.9679623+1.967962−13×−1.9679622−1=−1.341355

Substitute n=16 in xn+1=xn−xn3−xn−13xn2−1

x17=x16−x163−x16−13x162−1

Substitute x16=−1.341355 in x17=x16−x163−x16−13x162−1

x17=−1.341355−−1.3413553+1.341355−13×−1.3413552−1=−0.870187

Substitute n=17 in xn+1=xn−xn3−xn−13xn2−1

x18=x17−x173−x17−13x172−1

Substitute x17=−0.870187 in x18=x17−x173−x17−13x172−1

x18=−0.870187−−0.8701873+0.870187−13×−0.8701872−1=−0.249949

Substitute n=18 in xn+1=xn−xn3−xn−13xn2−1

x19=x18−x183−x18−13x182−1

Substitute x18=−0.249949 in x19=x18−x183−x18−13x182−1

x19=−0.249949−−0.2499493+0.249949−13×−0.2499492−1=−1.192219

Substitute n=19 in xn+1=xn−xn3−xn−13xn2−1

x20=x19−x193−x19−13x192−1

Substitute x19=−1.192219 in x20=x19−x193−x19−13x192−1

x20=−1.192219−−1.1922193+1.192219−13×−1.1922192−1=−0.731952

Substitute n=20 in xn+1=xn−xn3−xn−13xn2−1

x21=x20−x203−x20−13x202−1

Substitute x20=−0.731952 in x21=x20−x203−x20−13x202−1

x21=−0.731952−−0.7319523+0.731952−13×−0.7319522−1=0.355213

Substitute n=21 in xn+1=xn−xn3−xn−13xn2−1

x22=x21−x213−x21−13x212−1

Substitute x21=0.355213 in x22=x21−x213−x21−13x212−1

x22=0.355213−0.3552133−0.355213−13×0.3552132−1=−1.753322

Substitute n=22 in xn+1=xn−xn3−xn−13xn2−1

x23=x22−x223−x22−13x222−1

Substitute x22=−1.753322 in x23=x22−x223−x22−13x222−1

x23=(−1.753322)−(−1.753322)3−(−1.753322)−13(−1.753322)2−1=−1.189420

Substitute n=23 in xn+1=xn−xn3−xn−13xn2−1

x24=x23−x233−x23−13x232−1

Substitute x23=−1.189420 in x24=x23−x233−x23−13x232−1

x24=(−1.189420)−(−1.189420)3−(−1.189420)−13(−1.189420)2−1=−0.729123

Substitute n=24 in xn+1=xn−xn3−xn−13xn2−1

x25=x24−x243−x24−13x242−1

Substitute x24=−0.729123 in x25=x24−x243−x24−13x242−1,

x25=(−0.729123)−(−0.729123)3−(−0.729123)−13(−0.729123)2−1=0.377844

Substitute n=25 in xn+1=xn−xn3−xn−13xn2−1

x26=x25−x253−x25−13x252−1

Substitute x25=0.377844 in x26=x25−x253−x25−13x252−1,

x26=(0.377844)−(0.377844)3−(0.377844)−13(0.377844)2−1=−1.937872

Substitute n=26 in xn+1=xn−xn3−xn−13xn2−1

x27=x26−x263−x26−13x262−1

Substitute x26=−1.937872 in x27=x26−x263−x26−13x262−1,

x27=(−1.937872)−(−1.937872)3−(−1.937872)−13(−1.937872)2−1=−1.320350

Substitute n=27 in xn+1=xn−xn3−xn−13xn2−1

x28=x27−x273−x27−13x272−1

Substitute x27=−1.320350 in x28=x27−x273−x27−13x272−1,

x28=(−1.320350)−(−1.320350)3−(−1.320350)−13(−1.320350)2−1=−0.851919

Substitute n=28 in xn+1=xn−xn3−xn−13xn2−1

x29=x28−x283−x28−13x282−1

Substitute x28=−0.851919 in x29=x28−x283−x28−13x282−1,

x29=(−0.851919)−(−0.851919)3−(−0.851919)−13(−0.851919)2−1=−0.200959

Substitute n=29 in xn+1=xn−xn3−xn−13xn2−1

x30=x29−x293−x29−13x292−1

Substitute x29=−0.200959 in x30=x29−x293−x29−13x292−1,

Substitute n=30 in xn+1=xn−xn3−xn−13xn2−1

x31=x30−x303−x30−13x302−1

Substitute x30=−1.119386 in x31=x30−x303−x30−13x302−1

x31=−1.119386−−1.1193863+1.119386−13×−1.1193862−1=−0.654291

Substitute n=31 in xn+1=xn−xn3−xn−13xn2−1

x32=x31−x313−x31−13x312−1

Substitute x31=−0.654291 in x32=x31−x313−x31−13x312−1

x32=−0.654291−−0.6542913+0.654291−13×−0.6542912−1=1.547009

Substitute n=32 in xn+1=xn−xn3−xn−13xn2−1

x33=x32−x323−x32−13x322−1

Substitute x32=1.547009 in x33=x32−x323−x32−13x322−1

Substitute n=33 in xn+1=xn−xn3−xn−13xn2−1

x34=x33−x333−x33−13x332−1

Substitute x33=1.360050 in x34=x33−x333−x33−13x332−1

x34=1.360050−1.3600503−1.360050−13×1.3600502−1=1.325828

Substitute n=34 in xn+1=xn−xn3−xn−13xn2−1

x35=x34−x343−x34−13x342−1

Substitute x34=1.325828 in x35=x34−x343−x34−13x342−1

x35=1.325828−1.3258283−1.325828−13×1.3258282−1=1.324719

Substitute n=35 in xn+1=xn−xn3−xn−13xn2−1

x36=x35−x353−x35−13x352−1

Substitute x35=1.324719 in x36=x35−x353−x35−13x352−1

Substitute n=36 in xn+1=xn−xn3−xn−13xn2−1

x37=x36−x363−x36−13x362−1

Substitute x36=1.324718 in x37=x36−x363−x36−13x362−1

x37=1.324718−1.3247183−1.324718−13×1.3247182−1=1.324718

Here, x36=x37

Thus, the root of x is 1.324718.

(d)

Expert Solution
Check Mark
To determine

To sketch: the graph f(x)=x3−x−1 and its tangent lines at 1, 0.6, 0.57 to explain the dependency of initial approximation in Newton’s method.

Explanation of Solution

Given:

f(x)=x3−x−1

Calculation:

f(x)=x3−x−1

The graph is,

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.7, Problem 26E

In Figure 1,

The tangent line corresponding to x1=1 results in a sequence of approximations that converges quite quickly.

The tangent line corresponding to x1=0.6 is close to being horizontal, so x2 is quite far from the root. But the sequence converges slowly.

The tangent line corresponding to x1=0.57 is nearly horizontal, x2 is farther away from the root, and the sequence takes more iterations to converge.

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Chapter 4.7, Problem 25E
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Chapter 4.7, Problem 27E

Chapter 4 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Prob. 3ECh. 4.1 - Prob. 4ECh. 4.1 - Prob. 5ECh. 4.1 - Prob. 6ECh. 4.1 - Prob. 7ECh. 4.1 - Prob. 8ECh. 4.1 - Prob. 9ECh. 4.1 - Prob. 10E
Ch. 4.1 - Prob. 11ECh. 4.1 - Prob. 12ECh. 4.1 - Prob. 13ECh. 4.1 - Prob. 14ECh. 4.1 - Prob. 15ECh. 4.1 - Prob. 16ECh. 4.1 - Prob. 17ECh. 4.1 - Prob. 18ECh. 4.1 - Prob. 19ECh. 4.1 - Prob. 20ECh. 4.1 - Prob. 21ECh. 4.1 - Prob. 22ECh. 4.1 - Prob. 23ECh. 4.1 - Prob. 24ECh. 4.1 - Prob. 25ECh. 4.1 - Prob. 26ECh. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - Prob. 30ECh. 4.1 - Prob. 31ECh. 4.1 - Prob. 32ECh. 4.1 - Prob. 33ECh. 4.1 - Prob. 34ECh. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - Prob. 37ECh. 4.1 - Prob. 38ECh. 4.1 - Prob. 39ECh. 4.1 - Prob. 40ECh. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.2 - Explain the difference between an absolute minimum...Ch. 4.2 - Prob. 2ECh. 4.2 - Prob. 3ECh. 4.2 - For each of the numbers a, b, c, d, r, and s,...Ch. 4.2 - Prob. 5ECh. 4.2 - Use the graph to state the absolute and local...Ch. 4.2 - Prob. 7ECh. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - (a) Sketch the graph of a function that has a...Ch. 4.2 - Prob. 12ECh. 4.2 - (a) Sketch the graph of a function on [1, 2] that...Ch. 4.2 - Prob. 14ECh. 4.2 - Prob. 15ECh. 4.2 - Prob. 16ECh. 4.2 - Prob. 17ECh. 4.2 - Prob. 18ECh. 4.2 - Prob. 19ECh. 4.2 - Prob. 20ECh. 4.2 - Prob. 21ECh. 4.2 - Prob. 22ECh. 4.2 - Prob. 23ECh. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Prob. 26ECh. 4.2 - Find the critical numbers of the function. g(t) =...Ch. 4.2 - Prob. 28ECh. 4.2 - Find the critical numbers of the function....Ch. 4.2 - Prob. 30ECh. 4.2 - Prob. 31ECh. 4.2 - Prob. 32ECh. 4.2 - Prob. 33ECh. 4.2 - Find the critical numbers of the function. g() = 4...Ch. 4.2 - Find the critical numbers of the function. f() = 2...Ch. 4.2 - Find the critical numbers of the function. h(t) =...Ch. 4.2 - Find the critical numbers of the function. f(x) =...Ch. 4.2 - Prob. 38ECh. 4.2 - Prob. 39ECh. 4.2 - A formula for the derivative of a function f is...Ch. 4.2 - Prob. 41ECh. 4.2 - Prob. 42ECh. 4.2 - Prob. 43ECh. 4.2 - Prob. 44ECh. 4.2 - Prob. 45ECh. 4.2 - Prob. 46ECh. 4.2 - Prob. 47ECh. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Prob. 50ECh. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.2 - Prob. 53ECh. 4.2 - Prob. 54ECh. 4.2 - Prob. 55ECh. 4.2 - Prob. 56ECh. 4.2 - Prob. 57ECh. 4.2 - Prob. 58ECh. 4.2 - Prob. 59ECh. 4.2 - Prob. 60ECh. 4.2 - Prob. 61ECh. 4.2 - Prob. 62ECh. 4.2 - Prob. 63ECh. 4.2 - Prob. 64ECh. 4.2 - Prob. 65ECh. 4.2 - Prob. 66ECh. 4.3 - Prob. 1ECh. 4.3 - Prob. 2ECh. 4.3 - Prob. 3ECh. 4.3 - Prob. 4ECh. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - Prob. 8ECh. 4.3 - Prob. 9ECh. 4.3 - Prob. 10ECh. 4.3 - Prob. 11ECh. 4.3 - Prob. 12ECh. 4.3 - Prob. 13ECh. 4.3 - Prob. 14ECh. 4.3 - Prob. 15ECh. 4.3 - Prob. 16ECh. 4.3 - Prob. 17ECh. 4.3 - Prob. 18ECh. 4.3 - Prob. 19ECh. 4.3 - Prob. 20ECh. 4.3 - Prob. 21ECh. 4.3 - Prob. 22ECh. 4.3 - Prob. 23ECh. 4.3 - Prob. 24ECh. 4.3 - Prob. 25ECh. 4.3 - Prob. 26ECh. 4.3 - Prob. 27ECh. 4.3 - Prob. 28ECh. 4.3 - Prob. 29ECh. 4.3 - Prob. 30ECh. 4.3 - Prob. 31ECh. 4.3 - Prob. 32ECh. 4.3 - Prob. 33ECh. 4.3 - Prob. 34ECh. 4.3 - Prob. 35ECh. 4.3 - Prob. 36ECh. 4.3 - Prob. 37ECh. 4.3 - Prob. 38ECh. 4.3 - Prob. 39ECh. 4.3 - Prob. 40ECh. 4.3 - Prob. 41ECh. 4.3 - Prob. 42ECh. 4.3 - Prob. 43ECh. 4.3 - Prob. 44ECh. 4.3 - Prob. 45ECh. 4.3 - Prob. 46ECh. 4.3 - Prob. 47ECh. 4.3 - Prob. 48ECh. 4.3 - Prob. 49ECh. 4.3 - Prob. 50ECh. 4.3 - Prob. 51ECh. 4.3 - Prob. 52ECh. 4.3 - Prob. 53ECh. 4.3 - Prob. 54ECh. 4.3 - Prob. 55ECh. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.3 - Prob. 58ECh. 4.3 - Prob. 59ECh. 4.3 - Prob. 60ECh. 4.3 - Prob. 61ECh. 4.3 - Prob. 62ECh. 4.3 - Prob. 63ECh. 4.3 - Prob. 64ECh. 4.3 - Prob. 65ECh. 4.3 - Prob. 66ECh. 4.3 - Prob. 67ECh. 4.3 - Prob. 68ECh. 4.3 - Prob. 69ECh. 4.3 - Prob. 70ECh. 4.3 - Prob. 71ECh. 4.3 - Prob. 72ECh. 4.4 - Prob. 1ECh. 4.4 - Prob. 2ECh. 4.4 - Prob. 3ECh. 4.4 - Prob. 4ECh. 4.4 - Prob. 5ECh. 4.4 - Prob. 6ECh. 4.4 - Prob. 7ECh. 4.4 - Prob. 8ECh. 4.4 - Prob. 9ECh. 4.4 - Prob. 10ECh. 4.4 - Prob. 11ECh. 4.4 - Prob. 12ECh. 4.4 - Prob. 13ECh. 4.4 - Prob. 14ECh. 4.4 - Prob. 15ECh. 4.4 - Prob. 16ECh. 4.4 - Prob. 17ECh. 4.4 - Prob. 18ECh. 4.4 - Prob. 19ECh. 4.4 - Prob. 20ECh. 4.4 - Prob. 21ECh. 4.4 - Prob. 22ECh. 4.4 - Prob. 23ECh. 4.4 - Prob. 24ECh. 4.4 - Prob. 25ECh. 4.4 - Prob. 26ECh. 4.4 - Prob. 27ECh. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Prob. 30ECh. 4.4 - Prob. 31ECh. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.5 - Given that...Ch. 4.5 - Given that...Ch. 4.5 - Prob. 3ECh. 4.5 - Given that...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - Prob. 14ECh. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Prob. 19ECh. 4.5 - Prob. 20ECh. 4.5 - Prob. 21ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - Prob. 27ECh. 4.5 - Prob. 28ECh. 4.5 - Prob. 29ECh. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - Prob. 34ECh. 4.5 - Prob. 35ECh. 4.5 - Prob. 36ECh. 4.5 - Prob. 37ECh. 4.5 - Prob. 38ECh. 4.5 - Prob. 39ECh. 4.5 - Prob. 40ECh. 4.5 - Prob. 41ECh. 4.5 - Prob. 42ECh. 4.5 - Prob. 43ECh. 4.5 - Prob. 44ECh. 4.5 - Prob. 45ECh. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - Prob. 48ECh. 4.5 - Prob. 49ECh. 4.5 - Prob. 50ECh. 4.5 - Prob. 51ECh. 4.5 - Prob. 52ECh. 4.5 - Prob. 53ECh. 4.5 - Prob. 54ECh. 4.5 - Prob. 55ECh. 4.5 - Prob. 56ECh. 4.5 - Prob. 57ECh. 4.5 - Prob. 58ECh. 4.5 - Prob. 59ECh. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Prob. 62ECh. 4.5 - Prob. 63ECh. 4.5 - Prob. 64ECh. 4.5 - Prob. 65ECh. 4.5 - Prob. 66ECh. 4.5 - Prob. 67ECh. 4.5 - Prob. 68ECh. 4.5 - Prob. 69ECh. 4.5 - Prob. 70ECh. 4.5 - Prob. 71ECh. 4.5 - Prob. 72ECh. 4.5 - Prob. 73ECh. 4.5 - Prob. 74ECh. 4.5 - Prob. 75ECh. 4.5 - Prob. 76ECh. 4.6 - Consider the following problem: Find two numbers...Ch. 4.6 - Find two numbers whose difference is 100 and whose...Ch. 4.6 - Find two positive numbers whose product is 100 and...Ch. 4.6 - The sum of two positive numbers is 16. What is the...Ch. 4.6 - Prob. 5ECh. 4.6 - Prob. 6ECh. 4.6 - Prob. 7ECh. 4.6 - The rate (in mg carbon/m3/h) at which...Ch. 4.6 - Consider the following problem: A farmer with 750...Ch. 4.6 - Prob. 10ECh. 4.6 - Prob. 11ECh. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.6 - Prob. 14ECh. 4.6 - Prob. 15ECh. 4.6 - Prob. 16ECh. 4.6 - Prob. 17ECh. 4.6 - Prob. 18ECh. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Prob. 21ECh. 4.6 - Prob. 22ECh. 4.6 - Prob. 23ECh. 4.6 - Prob. 24ECh. 4.6 - Prob. 25ECh. 4.6 - Prob. 26ECh. 4.6 - Prob. 27ECh. 4.6 - Prob. 28ECh. 4.6 - Prob. 29ECh. 4.6 - Prob. 30ECh. 4.6 - Prob. 31ECh. 4.6 - Prob. 32ECh. 4.6 - Prob. 33ECh. 4.6 - Prob. 34ECh. 4.6 - Prob. 35ECh. 4.6 - Prob. 36ECh. 4.6 - Prob. 37ECh. 4.6 - Prob. 38ECh. 4.6 - Prob. 39ECh. 4.6 - Prob. 40ECh. 4.6 - Prob. 41ECh. 4.6 - Prob. 42ECh. 4.6 - Prob. 43ECh. 4.6 - Prob. 44ECh. 4.6 - Prob. 45ECh. 4.6 - Prob. 46ECh. 4.6 - Prob. 47ECh. 4.6 - Prob. 48ECh. 4.6 - Prob. 49ECh. 4.6 - Prob. 50ECh. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Prob. 53ECh. 4.6 - Prob. 54ECh. 4.6 - Prob. 55ECh. 4.6 - Prob. 56ECh. 4.6 - Prob. 57ECh. 4.6 - Prob. 58ECh. 4.6 - Prob. 59ECh. 4.6 - Prob. 60ECh. 4.6 - Prob. 61ECh. 4.6 - Prob. 62ECh. 4.7 - The figure shows the graph of a function f....Ch. 4.7 - Follow the instructions for Exercise 1(a) but use...Ch. 4.7 - Suppose the tangent line to the curve y = f(x) at...Ch. 4.7 - For each initial approximation, determine...Ch. 4.7 - Prob. 5ECh. 4.7 - Prob. 6ECh. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Use Newtons method with initial approximation x1 =...Ch. 4.7 - Use Newtons method with initial approximation x1 =...Ch. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - Prob. 13ECh. 4.7 - Prob. 14ECh. 4.7 - Prob. 15ECh. 4.7 - Prob. 16ECh. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - (a) Apply Newtons method to the equation x2 a = 0...Ch. 4.7 - (a) Apply Newtons method to the equation 1/x a =...Ch. 4.7 - (a) Use Newtons method with x1 = 1 to find the...Ch. 4.7 - Explain why Newtons method fails when applied to...Ch. 4.7 - Prob. 28ECh. 4.7 - Prob. 29ECh. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.8 - Prob. 1ECh. 4.8 - Prob. 2ECh. 4.8 - Prob. 3ECh. 4.8 - Prob. 4ECh. 4.8 - Prob. 5ECh. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - Prob. 8ECh. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - Prob. 16ECh. 4.8 - Prob. 19ECh. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Prob. 22ECh. 4.8 - Prob. 23ECh. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.8 - Prob. 30ECh. 4.8 - Prob. 31ECh. 4.8 - Prob. 32ECh. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Prob. 35ECh. 4.8 - Prob. 36ECh. 4.8 - Prob. 37ECh. 4.8 - Prob. 38ECh. 4.8 - The graph of f is shown in the figure. Sketch the...Ch. 4.8 - Prob. 40ECh. 4.8 - Prob. 41ECh. 4.8 - Prob. 42ECh. 4.8 - Prob. 43ECh. 4.8 - Prob. 44ECh. 4.8 - Prob. 45ECh. 4.8 - Prob. 46ECh. 4.8 - Prob. 47ECh. 4.8 - Prob. 48ECh. 4.8 - Prob. 49ECh. 4.8 - Prob. 50ECh. 4.8 - Prob. 51ECh. 4.8 - Prob. 52ECh. 4.8 - Prob. 53ECh. 4.8 - Prob. 54ECh. 4.8 - Prob. 55ECh. 4.8 - Prob. 56ECh. 4.8 - Prob. 57ECh. 4.8 - Prob. 58ECh. 4 - Prob. 1RCCCh. 4 - Prob. 2RCCCh. 4 - Prob. 3RCCCh. 4 - Prob. 4RCCCh. 4 - Prob. 5RCCCh. 4 - Prob. 6RCCCh. 4 - Prob. 7RCCCh. 4 - Prob. 8RCCCh. 4 - Prob. 9RCCCh. 4 - Prob. 10RCCCh. 4 - Prob. 1RQCh. 4 - Prob. 2RQCh. 4 - Prob. 3RQCh. 4 - Prob. 4RQCh. 4 - Prob. 5RQCh. 4 - Prob. 6RQCh. 4 - Prob. 7RQCh. 4 - Prob. 8RQCh. 4 - Prob. 9RQCh. 4 - Prob. 10RQCh. 4 - Prob. 11RQCh. 4 - Prob. 12RQCh. 4 - Prob. 13RQCh. 4 - If f and g are positive increasing functions on an...Ch. 4 - Prob. 15RQCh. 4 - Prob. 16RQCh. 4 - Prob. 17RQCh. 4 - Prob. 18RQCh. 4 - If f(x) exists and is nonzero for all x, then f(1)...Ch. 4 - limx0xex=1Ch. 4 - Prob. 1RECh. 4 - Prob. 2RECh. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Prob. 9RECh. 4 - Prob. 10RECh. 4 - Prob. 11RECh. 4 - Prob. 12RECh. 4 - Prob. 13RECh. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Prob. 17RECh. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Prob. 21RECh. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Prob. 27RECh. 4 - Prob. 28RECh. 4 - Prob. 29RECh. 4 - Prob. 30RECh. 4 - Prob. 31RECh. 4 - Prob. 32RECh. 4 - Prob. 33RECh. 4 - Prob. 34RECh. 4 - Prob. 35RECh. 4 - Prob. 36RECh. 4 - Prob. 37RECh. 4 - Prob. 38RECh. 4 - Prob. 39RECh. 4 - Prob. 40RECh. 4 - Prob. 41RECh. 4 - Prob. 42RECh. 4 - Prob. 43RECh. 4 - Prob. 44RECh. 4 - Prob. 45RECh. 4 - Prob. 46RECh. 4 - Prob. 47RECh. 4 - Prob. 48RECh. 4 - Prob. 49RECh. 4 - Prob. 50RECh. 4 - Prob. 51RECh. 4 - Prob. 52RECh. 4 - Prob. 53RECh. 4 - Prob. 54RECh. 4 - Prob. 55RECh. 4 - Prob. 56RECh. 4 - Prob. 57RECh. 4 - Prob. 58RECh. 4 - Prob. 59RECh. 4 - Prob. 60RECh. 4 - Prob. 61RECh. 4 - Prob. 62RECh. 4 - Prob. 63RECh. 4 - Prob. 64RECh. 4 - Prob. 65RECh. 4 - If a rectangle has its base on the x-axis and two...Ch. 4 - Show that sinxcosx2 for all x.Ch. 4 - Prob. 3PCh. 4 - Prob. 4PCh. 4 - Prob. 5PCh. 4 - Find the point on the parabola y = 1 x2 at which...Ch. 4 - Prob. 7PCh. 4 - Prob. 8PCh. 4 - Prob. 9PCh. 4 - Prob. 10PCh. 4 - Prob. 11PCh. 4 - Prob. 12PCh. 4 - Prob. 13PCh. 4 - Prob. 14PCh. 4 - Prob. 15PCh. 4 - Prob. 16PCh. 4 - Prob. 17PCh. 4 - Prob. 18PCh. 4 - Prob. 19PCh. 4 - Prob. 20PCh. 4 - Prob. 21PCh. 4 - Prob. 22PCh. 4 - Prob. 23PCh. 4 - Prob. 24P
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